Statistical Mechanics

Question	Answer
What is $\Omega$ for $k$ indistinguishable elements from $n$ objects with replacement?	$\Omega=n^k$
What is $\Omega$ for $k$ distinguishable elements from $n$ objects without replacement?	$\Omega=\frac{n!}{(n-k)!}$
What is $\Omega$ for $k$ indistinguishable elements from $n$ objects without replacement?	$\Omega={n\choose k}=\frac{n!}{k!(n-k)!}$
Sterling Approximation	$ln(N!)=N\,ln(N)-N+O(ln(N))$
The Microcanonical Interconnect between Statistical Mechanics and Thermodynamics	Entropy: $S=k_bln(\Omega)$
The Canonical Partition Function for Classical Systems	$Z_{CM}=\frac{1}{h^N}\int\cdots\int\prod_{i=1}^Ndp_idq_ie^{-H(p,q)\beta}$
Microcanonical Partition Function	$Z=\sum_j\omega_je^{\beta E_j}$
Canonical Partition Function	$Z_{can}(N,V,T)=\frac{1}{N!}\frac{1}{h^{3N}}\int\prod_{i=1}^{3N}dp_idq_ie^{-H(p,q)\beta}$
The Canonical Interconnect between Statistical Mechanics and Thermodynamics	Helmholtz Free Energy: $F(N,V,T)=-kT\,ln(Z_{can}(N,V,T))$
The Grand Canonical Partition Function	$Z_G=e^{\lambda Z_{can}}$ where $\lambda=e^{\mu/kT}$ is the "fugacity"
The Grand Canonical Interconnect between Statistical Mechanics and Thermodynamics	The Grand Potential: $\Phi(T,V,\mu)=-pV=-kT\,ln(Z_G)$
Characteristic of Microcanonical Ensembles	Every state is equally probable.
Characteristic of Canonical Ensembles	Energy is allowed to fluctuate.
Characteristic of Grand Canonical Ensembles	Energy and particle number are allowed to fluctuate.
The Canonical Equation for Entropy	$S=\frac{\partial}{\partial T}(T\,ln(Z_{can}))$
The Canonical Equation for Internal Energy	$U=k_bT^2\frac{\partial}{\partial T}ln(Z_{can})$
The Grand Canonical Equation for Total Particle Number	$N=k_bT\frac{\partial}{\partial\mu}ln(Z_G)$
The Grand Canonical Equation for Fluctuations in Particle Number	$(\Delta N)^2=\frac{\partial^2}{\partial(\mu\beta)^2}ln(Z_G)$
/[dE/]	/[dE=T\;dS-P\;dV+\mu\;dN/]
$dF$	$dF=-S\;dT-P\;dV+\mu\;dN$ $F=E-T\,S$
$C_V$	$T\frac{\partial S}{\partial T}\Big\|_{V,N}=\frac{\partial E}{\partial T}\Big\|_{V,N}$

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	Created by Kate E about 5 years ago